1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

382 13. WEP and LLP

Lemma 13.3.5. Let A and B be unital C* -algebras. If B has the WEP and

there exists a u.c.p. map cp: B ---+A which maps the closed unit ball of B

onto the closed unit ball of A, then A is QWEP.

Proof. Let Bo c B be the multiplicative domain of cp (Definition 1.5.8).

Since cp maps the closed unit ball of B onto that of A, the restriction of

cp to Bo is a surjective *-homomorphism 7r onto A. We will prove that

Bo has the WEP. By Proposition 3.6.6, it suffices to show the canonical

*-homomorphism Bo ®max C---+ B ®max C is injective for every C*-algebra

C. Let J = {x EB: cp(x*x) = 0 = cp(xx*)} be a hereditary C*-subalgebra

in B. We observe that J c Bo and in fact J = ker7r. Injectivity follows

from the commutative diagram

'll"@id

O ~ J ®max C~ Bo ®max C ~ A®max C ~ O

II l ~@id II

J ®max Cc_______,,.. B ®max C _____,.. A ®max C

since the top row is exact and J ®max C---+ B ®max C is injective (since J is

hereditary). D

Lemma 13.3.6. Let {AihEI be an increasing net ofC*-subalgebras inIIB(1i).

If the Ai 's are all Q WEP, then so are LJ A and (LJ Ai)''.

Proof. We only prove. that the von Neumann algebra M = (LJ Ai)" is
QWEP. Adjoining the unit of JIB(7i) if necessary, we may assume that all
Ai's are unital. For each i E I, fix a C-algebra Bi with the WEP and a
surjective -homomorphism 7ri: Bi ---+ Ai. Let J be a suitable directed set.
It follows from Lemma 13.3.3 that B = IT(i,j)EixJ Bi has the WEP. Fix a
cofinal ultrafilter U (resp. V) on the directed set I (resp. J) and define a
u.c.p. map cp: B ---+ M by
cp((x· i,J ·)) = j-+Vi-+U lim lim 1f·(x· i i,J ·) '

where the limits are taken with respect to the ultraweak topology. If J is

chosen large enough, then cp maps the closed unit ball of B onto that of M,

by Kaplansky's density theorem. By Lemma 13.3.5, this implies that M is

QWEP. D

Every C* -algebra A is obviously relatively weakly injective in its double
dual A. Hence Lemmas 13.3.4 and 13.3.6 imply the useful fact that A is
Q WEP if and only if A is Q WEP.

Proof of Theorem 13.3.1. (1) =? (2): It suffices to show that if a C-
algebra A is QWEP and has the LP, then A has the WEP. Let B be a C -
algebra with the WEP and let 7r: B ---+ A be a surjective *-homomorphism.